Impact of Doubling the Plate Separation on Energy Stored in a Parallel Plate Capacitor

When a parallel plate capacitor is connected to a battery, the voltage across the capacitor remains constant. If the plate separation is doubled, several interesting changes occur, particularly in the capacitance, the charge stored, and the energy stored in the capacitor.

Capacitance Change

The capacitance C of a parallel plate capacitor is given by the formula:

C frac{varepsilon_0 A}{d}

where (varepsilon_0) is the permittivity of free space, A is the area of the plates, and d is the separation between the plates. If the separation d is doubled, the new capacitance C' becomes:

C' frac{varepsilon_0 A}{2d} frac{C}{2}

This means the capacitance is halved. When the plate separation is doubled, the capacitance is reduced to half of its original value.

Charge Change

The charge Q stored in a capacitor is given by:

Q C cdot V

Since the capacitor remains connected to the battery, the voltage V remains constant. When the capacitance is halved, the new charge Q' becomes:

Q' C cdot V left(frac{C}{2}right) cdot V frac{Q}{2}

Thus, the charge on the capacitor is halved. The electric field is the same as the voltage remains constant, but the plate separation increases, reducing the ability of the capacitor to store charge.

Stored Energy Change

The energy U stored in a capacitor is given by:

U frac{1}{2} C V^2

With the new capacitance, the new energy U' becomes:

U' frac{1}{2} C V^2 frac{1}{2} left(frac{C}{2}right) V^2 frac{1}{4} C V^2 frac{U}{2}

This means that the energy stored in the capacitor is also reduced to half of its original value.

Summary

The behavior of the parallel plate capacitor illustrates how the energy stored in a capacitor is directly related to both its capacitance and the voltage across it. When the plate separation is doubled, the capacitance, charge, and stored energy are all halved.

This relationship can be summarized as follows:

The capacitance is halved. The charge stored on the capacitor is halved. The energy stored in the capacitor is also halved.

Understanding these effects is crucial for optimizing the performance of capacitors in various electronic and electrical systems. Proper design considerations are essential to maintain desired operational parameters.